climleltrp - Mathbox for Glauco Siliprandi

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Theorem climleltrp 42318

Description: The limit of complex number sequence 𝐹 is eventually approximated. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

**Hypotheses**
Ref	Expression
climleltrp.k	⊢ Ⅎ𝑘𝜑
climleltrp.f	⊢ Ⅎ𝑘𝐹
climleltrp.z	⊢ 𝑍 = (ℤ_≥‘𝑀)
climleltrp.n	⊢ (𝜑 → 𝑁 ∈ 𝑍)
climleltrp.r	⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑘) ∈ ℝ)
climleltrp.a	⊢ (𝜑 → 𝐹 ⇝ 𝐴)
climleltrp.c	⊢ (𝜑 → 𝐶 ∈ ℝ)
climleltrp.l	⊢ (𝜑 → 𝐴 ≤ 𝐶)
climleltrp.x	⊢ (𝜑 → 𝑋 ∈ ℝ⁺)

**Assertion**
Ref	Expression
climleltrp	⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋)))

Distinct variable groups: 𝐴,𝑗,𝑘 𝑗,𝐹 𝑗,𝑁,𝑘 𝑗,𝑋,𝑘 𝑗,𝑍 𝜑,𝑗 Allowed substitution hints: 𝜑(𝑘) 𝐶(𝑗,𝑘) 𝐹(𝑘) 𝑀(𝑗,𝑘) 𝑍(𝑘)

**Proof of Theorem climleltrp**
Step	Hyp	Ref	Expression
1		climleltrp.n	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ 𝑍)
2		climleltrp.z	. . . . 5 ⊢ 𝑍 = (ℤ_≥‘𝑀)
3	1, 2	eleqtrdi 2900	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
4		uzss 12253	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝑁) ⊆ (ℤ_≥‘𝑀))
5	3, 4	syl 17	. . 3 ⊢ (𝜑 → (ℤ_≥‘𝑁) ⊆ (ℤ_≥‘𝑀))
6	5, 2	sseqtrrdi 3966	. 2 ⊢ (𝜑 → (ℤ_≥‘𝑁) ⊆ 𝑍)
7		climleltrp.k	. . . 4 ⊢ Ⅎ𝑘𝜑
8		climleltrp.f	. . . 4 ⊢ Ⅎ𝑘𝐹
9		uzssz 12252	. . . . 5 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
10	9, 3	sseldi 3913	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ)
11		eqid 2798	. . . 4 ⊢ (ℤ_≥‘𝑁) = (ℤ_≥‘𝑁)
12		climleltrp.a	. . . 4 ⊢ (𝜑 → 𝐹 ⇝ 𝐴)
13		eqidd 2799	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑘) = (𝐹‘𝑘))
14		climleltrp.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ℝ⁺)
15	7, 8, 10, 11, 12, 13, 14	clim2d 42315	. . 3 ⊢ (𝜑 → ∃𝑗 ∈ (ℤ_≥‘𝑁)∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋))
16		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑘 𝑗 ∈ (ℤ_≥‘𝑁)
17	7, 16	nfan 1900	. . . . 5 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁))
18		simplll 774	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → 𝜑)
19		uzss 12253	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (ℤ_≥‘𝑁) → (ℤ_≥‘𝑗) ⊆ (ℤ_≥‘𝑁))
20	19	ad2antlr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (ℤ_≥‘𝑗) ⊆ (ℤ_≥‘𝑁))
21		simpr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ (ℤ_≥‘𝑗))
22	20, 21	sseldd 3916	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ (ℤ_≥‘𝑁))
23	22	adantr 484	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → 𝑘 ∈ (ℤ_≥‘𝑁))
24		simpr 488	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋)
25		climleltrp.r	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑘) ∈ ℝ)
26	13, 25	eqeltrd 2890	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑘) ∈ ℝ)
27	26	adantr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (𝐹‘𝑘) ∈ ℝ)
28		climcl 14848	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ)
29	12, 28	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ ℂ)
30	29	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → 𝐴 ∈ ℂ)
31	26	recnd 10658	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑘) ∈ ℂ)
32	30, 31	pncan3d 10989	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → (𝐴 + ((𝐹‘𝑘) − 𝐴)) = (𝐹‘𝑘))
33	32	eqcomd 2804	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑘) = (𝐴 + ((𝐹‘𝑘) − 𝐴)))
34	33	adantr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (𝐹‘𝑘) = (𝐴 + ((𝐹‘𝑘) − 𝐴)))
35	34, 27	eqeltrrd 2891	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (𝐴 + ((𝐹‘𝑘) − 𝐴)) ∈ ℝ)
36		climleltrp.c	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ ℝ)
37	36	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → 𝐶 ∈ ℝ)
38	7, 8, 11, 10, 12, 25	climreclf 42306	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ ℝ)
39	38	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → 𝐴 ∈ ℝ)
40	27, 39	resubcld 11057	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → ((𝐹‘𝑘) − 𝐴) ∈ ℝ)
41	37, 40	readdcld 10659	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (𝐶 + ((𝐹‘𝑘) − 𝐴)) ∈ ℝ)
42	14	rpred 12419	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ∈ ℝ)
43	42	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → 𝑋 ∈ ℝ)
44	37, 43	readdcld 10659	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (𝐶 + 𝑋) ∈ ℝ)
45		climleltrp.l	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ≤ 𝐶)
46	45	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → 𝐴 ≤ 𝐶)
47	39, 37, 40, 46	leadd1dd 11243	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (𝐴 + ((𝐹‘𝑘) − 𝐴)) ≤ (𝐶 + ((𝐹‘𝑘) − 𝐴)))
48	31	adantr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (𝐹‘𝑘) ∈ ℂ)
49	30	adantr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → 𝐴 ∈ ℂ)
50	48, 49	subcld 10986	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → ((𝐹‘𝑘) − 𝐴) ∈ ℂ)
51	50	abscld 14788	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (abs‘((𝐹‘𝑘) − 𝐴)) ∈ ℝ)
52	40	leabsd 14766	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → ((𝐹‘𝑘) − 𝐴) ≤ (abs‘((𝐹‘𝑘) − 𝐴)))
53		simpr 488	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋)
54	40, 51, 43, 52, 53	lelttrd 10787	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → ((𝐹‘𝑘) − 𝐴) < 𝑋)
55	40, 43, 37, 54	ltadd2dd 10788	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (𝐶 + ((𝐹‘𝑘) − 𝐴)) < (𝐶 + 𝑋))
56	35, 41, 44, 47, 55	lelttrd 10787	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (𝐴 + ((𝐹‘𝑘) − 𝐴)) < (𝐶 + 𝑋))
57	34, 56	eqbrtrd 5052	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (𝐹‘𝑘) < (𝐶 + 𝑋))
58	27, 57	jca 515	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → ((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋)))
59	18, 23, 24, 58	syl21anc 836	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → ((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋)))
60	59	adantrl 715	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋)) → ((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋)))
61	60	ex 416	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → ((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋))))
62	17, 61	ralimdaa 3181	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → (∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋))))
63	62	reximdva 3233	. . 3 ⊢ (𝜑 → (∃𝑗 ∈ (ℤ_≥‘𝑁)∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → ∃𝑗 ∈ (ℤ_≥‘𝑁)∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋))))
64	15, 63	mpd 15	. 2 ⊢ (𝜑 → ∃𝑗 ∈ (ℤ_≥‘𝑁)∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋)))
65		ssrexv 3982	. 2 ⊢ ((ℤ_≥‘𝑁) ⊆ 𝑍 → (∃𝑗 ∈ (ℤ_≥‘𝑁)∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋)) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋))))
66	6, 64, 65	sylc 65	1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 Ⅎwnf 1785 ∈ wcel 2111 Ⅎwnfc 2936 ∀wral 3106 ∃wrex 3107 ⊆ wss 3881 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 + caddc 10529 < clt 10664 ≤ cle 10665 − cmin 10859 ℤcz 11969 ℤ_≥cuz 12231 ℝ⁺crp 12377 abscabs 14585 ⇝ cli 14833

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fl 13157 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-rlim 14838

This theorem is referenced by: smflimlem2 43405

W3C validator